Joint continuity of the local times of fractional brownian sheets

Antoine Ayache; Dongsheng Wu; Yimin Xiao

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 4, page 727-748
  • ISSN: 0246-0203

Abstract

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Let BH={BH(t), t∈ℝ+N} be an (N, d)-fractional brownian sheet with index H=(H1, …, HN)∈(0, 1)N defined by BH(t)=(BH1(t), …, BHd(t)) (t∈ℝ+N), where BH1, …, BHd are independent copies of a real-valued fractional brownian sheet B0H. We prove that if d<∑ℓ=1NHℓ−1, then the local times of BH are jointly continuous. This verifies a conjecture of Xiao and Zhang (Probab. Theory Related Fields124 (2002)). We also establish sharp local and global Hölder conditions for the local times of BH. These results are applied to study analytic and geometric properties of the sample paths of BH.

How to cite

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Ayache, Antoine, Wu, Dongsheng, and Xiao, Yimin. "Joint continuity of the local times of fractional brownian sheets." Annales de l'I.H.P. Probabilités et statistiques 44.4 (2008): 727-748. <http://eudml.org/doc/77989>.

@article{Ayache2008,
abstract = {Let BH=\{BH(t), t∈ℝ+N\} be an (N, d)-fractional brownian sheet with index H=(H1, …, HN)∈(0, 1)N defined by BH(t)=(BH1(t), …, BHd(t)) (t∈ℝ+N), where BH1, …, BHd are independent copies of a real-valued fractional brownian sheet B0H. We prove that if d&lt;∑ℓ=1NHℓ−1, then the local times of BH are jointly continuous. This verifies a conjecture of Xiao and Zhang (Probab. Theory Related Fields124 (2002)). We also establish sharp local and global Hölder conditions for the local times of BH. These results are applied to study analytic and geometric properties of the sample paths of BH.},
author = {Ayache, Antoine, Wu, Dongsheng, Xiao, Yimin},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {fractional brownian sheet; Liouville fractional brownian sheet; fractional brownian motion; sectorial local nondeterminism; local times; joint continuity; Hölder conditions; fractional Brownian sheet; Liouville fractional Brownian sheet; fractional Brownian motion},
language = {eng},
number = {4},
pages = {727-748},
publisher = {Gauthier-Villars},
title = {Joint continuity of the local times of fractional brownian sheets},
url = {http://eudml.org/doc/77989},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Ayache, Antoine
AU - Wu, Dongsheng
AU - Xiao, Yimin
TI - Joint continuity of the local times of fractional brownian sheets
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 4
SP - 727
EP - 748
AB - Let BH={BH(t), t∈ℝ+N} be an (N, d)-fractional brownian sheet with index H=(H1, …, HN)∈(0, 1)N defined by BH(t)=(BH1(t), …, BHd(t)) (t∈ℝ+N), where BH1, …, BHd are independent copies of a real-valued fractional brownian sheet B0H. We prove that if d&lt;∑ℓ=1NHℓ−1, then the local times of BH are jointly continuous. This verifies a conjecture of Xiao and Zhang (Probab. Theory Related Fields124 (2002)). We also establish sharp local and global Hölder conditions for the local times of BH. These results are applied to study analytic and geometric properties of the sample paths of BH.
LA - eng
KW - fractional brownian sheet; Liouville fractional brownian sheet; fractional brownian motion; sectorial local nondeterminism; local times; joint continuity; Hölder conditions; fractional Brownian sheet; Liouville fractional Brownian sheet; fractional Brownian motion
UR - http://eudml.org/doc/77989
ER -

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